LMFDB - Embedded newform 8006.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8006,2,Mod(1,8006)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8006 = 2 \cdot 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8006.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9282318582$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8006.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.28301 q^{3} +1.00000 q^{4} -1.18461 q^{5} -3.28301 q^{6} +0.314789 q^{7} +1.00000 q^{8} +7.77813 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.28301 q^{3} +1.00000 q^{4} -1.18461 q^{5} -3.28301 q^{6} +0.314789 q^{7} +1.00000 q^{8} +7.77813 q^{9} -1.18461 q^{10} -1.55638 q^{11} -3.28301 q^{12} +2.14208 q^{13} +0.314789 q^{14} +3.88907 q^{15} +1.00000 q^{16} -3.52465 q^{17} +7.77813 q^{18} +1.78556 q^{19} -1.18461 q^{20} -1.03346 q^{21} -1.55638 q^{22} +4.34340 q^{23} -3.28301 q^{24} -3.59671 q^{25} +2.14208 q^{26} -15.6866 q^{27} +0.314789 q^{28} -4.40062 q^{29} +3.88907 q^{30} -3.94235 q^{31} +1.00000 q^{32} +5.10960 q^{33} -3.52465 q^{34} -0.372901 q^{35} +7.77813 q^{36} +4.98956 q^{37} +1.78556 q^{38} -7.03245 q^{39} -1.18461 q^{40} +10.1007 q^{41} -1.03346 q^{42} -7.20934 q^{43} -1.55638 q^{44} -9.21401 q^{45} +4.34340 q^{46} -2.85379 q^{47} -3.28301 q^{48} -6.90091 q^{49} -3.59671 q^{50} +11.5715 q^{51} +2.14208 q^{52} +0.974012 q^{53} -15.6866 q^{54} +1.84370 q^{55} +0.314789 q^{56} -5.86199 q^{57} -4.40062 q^{58} +3.56507 q^{59} +3.88907 q^{60} +10.6858 q^{61} -3.94235 q^{62} +2.44847 q^{63} +1.00000 q^{64} -2.53752 q^{65} +5.10960 q^{66} +2.19986 q^{67} -3.52465 q^{68} -14.2594 q^{69} -0.372901 q^{70} -14.5629 q^{71} +7.77813 q^{72} +1.87294 q^{73} +4.98956 q^{74} +11.8080 q^{75} +1.78556 q^{76} -0.489931 q^{77} -7.03245 q^{78} -5.95226 q^{79} -1.18461 q^{80} +28.1649 q^{81} +10.1007 q^{82} +0.696817 q^{83} -1.03346 q^{84} +4.17533 q^{85} -7.20934 q^{86} +14.4472 q^{87} -1.55638 q^{88} +9.21616 q^{89} -9.21401 q^{90} +0.674303 q^{91} +4.34340 q^{92} +12.9427 q^{93} -2.85379 q^{94} -2.11518 q^{95} -3.28301 q^{96} +6.15074 q^{97} -6.90091 q^{98} -12.1057 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9} - 9 q^{10} - 48 q^{11} - 15 q^{12} - 30 q^{13} - 29 q^{14} - 51 q^{15} + 69 q^{16} - 37 q^{17} + 40 q^{18} - 72 q^{19} - 9 q^{20} - 38 q^{21} - 48 q^{22} - 75 q^{23} - 15 q^{24} + 18 q^{25} - 30 q^{26} - 48 q^{27} - 29 q^{28} - 27 q^{29} - 51 q^{30} - 61 q^{31} + 69 q^{32} - 29 q^{33} - 37 q^{34} - 64 q^{35} + 40 q^{36} - 42 q^{37} - 72 q^{38} - 68 q^{39} - 9 q^{40} - 49 q^{41} - 38 q^{42} - 95 q^{43} - 48 q^{44} - 20 q^{45} - 75 q^{46} - 62 q^{47} - 15 q^{48} - 4 q^{49} + 18 q^{50} - 76 q^{51} - 30 q^{52} - 28 q^{53} - 48 q^{54} - 76 q^{55} - 29 q^{56} - 44 q^{57} - 27 q^{58} - 68 q^{59} - 51 q^{60} - 62 q^{61} - 61 q^{62} - 91 q^{63} + 69 q^{64} - 79 q^{65} - 29 q^{66} - 116 q^{67} - 37 q^{68} - 23 q^{69} - 64 q^{70} - 89 q^{71} + 40 q^{72} - 60 q^{73} - 42 q^{74} - 47 q^{75} - 72 q^{76} + 5 q^{77} - 68 q^{78} - 170 q^{79} - 9 q^{80} - 3 q^{81} - 49 q^{82} - 82 q^{83} - 38 q^{84} - 81 q^{85} - 95 q^{86} - 51 q^{87} - 48 q^{88} - 78 q^{89} - 20 q^{90} - 85 q^{91} - 75 q^{92} - 21 q^{93} - 62 q^{94} - 70 q^{95} - 15 q^{96} - 60 q^{97} - 4 q^{98} - 148 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9 - 9 * q^10 - 48 * q^11 - 15 * q^12 - 30 * q^13 - 29 * q^14 - 51 * q^15 + 69 * q^16 - 37 * q^17 + 40 * q^18 - 72 * q^19 - 9 * q^20 - 38 * q^21 - 48 * q^22 - 75 * q^23 - 15 * q^24 + 18 * q^25 - 30 * q^26 - 48 * q^27 - 29 * q^28 - 27 * q^29 - 51 * q^30 - 61 * q^31 + 69 * q^32 - 29 * q^33 - 37 * q^34 - 64 * q^35 + 40 * q^36 - 42 * q^37 - 72 * q^38 - 68 * q^39 - 9 * q^40 - 49 * q^41 - 38 * q^42 - 95 * q^43 - 48 * q^44 - 20 * q^45 - 75 * q^46 - 62 * q^47 - 15 * q^48 - 4 * q^49 + 18 * q^50 - 76 * q^51 - 30 * q^52 - 28 * q^53 - 48 * q^54 - 76 * q^55 - 29 * q^56 - 44 * q^57 - 27 * q^58 - 68 * q^59 - 51 * q^60 - 62 * q^61 - 61 * q^62 - 91 * q^63 + 69 * q^64 - 79 * q^65 - 29 * q^66 - 116 * q^67 - 37 * q^68 - 23 * q^69 - 64 * q^70 - 89 * q^71 + 40 * q^72 - 60 * q^73 - 42 * q^74 - 47 * q^75 - 72 * q^76 + 5 * q^77 - 68 * q^78 - 170 * q^79 - 9 * q^80 - 3 * q^81 - 49 * q^82 - 82 * q^83 - 38 * q^84 - 81 * q^85 - 95 * q^86 - 51 * q^87 - 48 * q^88 - 78 * q^89 - 20 * q^90 - 85 * q^91 - 75 * q^92 - 21 * q^93 - 62 * q^94 - 70 * q^95 - 15 * q^96 - 60 * q^97 - 4 * q^98 - 148 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.28301	−1.89544	−0.947722	−	0.319097i	$-0.896621\pi$
$3$	−3.28301	−1.89544	−0.947722	+	0.319097i	$0.896621\pi$
$4$	1.00000	0.500000
$5$	−1.18461	−0.529772	−0.264886	−	0.964280i	$-0.585334\pi$
$5$	−1.18461	−0.529772	−0.264886	+	0.964280i	$0.585334\pi$
$6$	−3.28301	−1.34028
$7$	0.314789	0.118979	0.0594896	−	0.998229i	$-0.481053\pi$
$7$	0.314789	0.118979	0.0594896	+	0.998229i	$0.481053\pi$
$8$	1.00000	0.353553
$9$	7.77813	2.59271
$10$	−1.18461	−0.374605
$11$	−1.55638	−0.469266	−0.234633	−	0.972084i	$-0.575389\pi$
$11$	−1.55638	−0.469266	−0.234633	+	0.972084i	$0.575389\pi$
$12$	−3.28301	−0.947722
$13$	2.14208	0.594105	0.297053	−	0.954861i	$-0.403996\pi$
$13$	2.14208	0.594105	0.297053	+	0.954861i	$0.403996\pi$
$14$	0.314789	0.0841310
$15$	3.88907	1.00415
$16$	1.00000	0.250000
$17$	−3.52465	−0.854854	−0.427427	−	0.904050i	$-0.640580\pi$
$17$	−3.52465	−0.854854	−0.427427	+	0.904050i	$0.640580\pi$
$18$	7.77813	1.83332
$19$	1.78556	0.409634	0.204817	−	0.978800i	$-0.434340\pi$
$19$	1.78556	0.409634	0.204817	+	0.978800i	$0.434340\pi$
$20$	−1.18461	−0.264886
$21$	−1.03346	−0.225518
$22$	−1.55638	−0.331821
$23$	4.34340	0.905662	0.452831	−	0.891596i	$-0.350414\pi$
$23$	4.34340	0.905662	0.452831	+	0.891596i	$0.350414\pi$
$24$	−3.28301	−0.670141
$25$	−3.59671	−0.719342
$26$	2.14208	0.420096
$27$	−15.6866	−3.01889
$28$	0.314789	0.0594896
$29$	−4.40062	−0.817174	−0.408587	−	0.912719i	$-0.633978\pi$
$29$	−4.40062	−0.817174	−0.408587	+	0.912719i	$0.633978\pi$
$30$	3.88907	0.710043
$31$	−3.94235	−0.708066	−0.354033	−	0.935233i	$-0.615190\pi$
$31$	−3.94235	−0.708066	−0.354033	+	0.935233i	$0.615190\pi$
$32$	1.00000	0.176777
$33$	5.10960	0.889467
$34$	−3.52465	−0.604473
$35$	−0.372901	−0.0630318
$36$	7.77813	1.29635
$37$	4.98956	0.820279	0.410139	−	0.912023i	$-0.365480\pi$
$37$	4.98956	0.820279	0.410139	+	0.912023i	$0.365480\pi$
$38$	1.78556	0.289655
$39$	−7.03245	−1.12609
$40$	−1.18461	−0.187303
$41$	10.1007	1.57746	0.788731	−	0.614738i	$-0.210738\pi$
$41$	10.1007	1.57746	0.788731	+	0.614738i	$0.210738\pi$
$42$	−1.03346	−0.159466
$43$	−7.20934	−1.09941	−0.549707	−	0.835357i	$-0.685261\pi$
$43$	−7.20934	−1.09941	−0.549707	+	0.835357i	$0.685261\pi$
$44$	−1.55638	−0.234633
$45$	−9.21401	−1.37354
$46$	4.34340	0.640400
$47$	−2.85379	−0.416268	−0.208134	−	0.978100i	$-0.566739\pi$
$47$	−2.85379	−0.416268	−0.208134	+	0.978100i	$0.566739\pi$
$48$	−3.28301	−0.473861
$49$	−6.90091	−0.985844
$50$	−3.59671	−0.508652
$51$	11.5715	1.62033
$52$	2.14208	0.297053
$53$	0.974012	0.133791	0.0668954	−	0.997760i	$-0.478691\pi$
$53$	0.974012	0.133791	0.0668954	+	0.997760i	$0.478691\pi$
$54$	−15.6866	−2.13468
$55$	1.84370	0.248604
$56$	0.314789	0.0420655
$57$	−5.86199	−0.776439
$58$	−4.40062	−0.577829
$59$	3.56507	0.464132	0.232066	−	0.972700i	$-0.425451\pi$
$59$	3.56507	0.464132	0.232066	+	0.972700i	$0.425451\pi$
$60$	3.88907	0.502076
$61$	10.6858	1.36818	0.684088	−	0.729399i	$-0.260200\pi$
$61$	10.6858	1.36818	0.684088	+	0.729399i	$0.260200\pi$
$62$	−3.94235	−0.500678
$63$	2.44847	0.308478
$64$	1.00000	0.125000
$65$	−2.53752	−0.314740
$66$	5.10960	0.628948
$67$	2.19986	0.268755	0.134378	−	0.990930i	$-0.457096\pi$
$67$	2.19986	0.268755	0.134378	+	0.990930i	$0.457096\pi$
$68$	−3.52465	−0.427427
$69$	−14.2594	−1.71663
$70$	−0.372901	−0.0445702
$71$	−14.5629	−1.72829	−0.864147	−	0.503240i	$-0.832141\pi$
$71$	−14.5629	−1.72829	−0.864147	+	0.503240i	$0.832141\pi$
$72$	7.77813	0.916661
$73$	1.87294	0.219211	0.109605	−	0.993975i	$-0.465041\pi$
$73$	1.87294	0.219211	0.109605	+	0.993975i	$0.465041\pi$
$74$	4.98956	0.580025
$75$	11.8080	1.36347
$76$	1.78556	0.204817
$77$	−0.489931	−0.0558329
$78$	−7.03245	−0.796268
$79$	−5.95226	−0.669682	−0.334841	−	0.942275i	$-0.608683\pi$
$79$	−5.95226	−0.669682	−0.334841	+	0.942275i	$0.608683\pi$
$80$	−1.18461	−0.132443
$81$	28.1649	3.12943
$82$	10.1007	1.11543
$83$	0.696817	0.0764857	0.0382428	−	0.999268i	$-0.487824\pi$
$83$	0.696817	0.0764857	0.0382428	+	0.999268i	$0.487824\pi$
$84$	−1.03346	−0.112759
$85$	4.17533	0.452878
$86$	−7.20934	−0.777404
$87$	14.4472	1.54891
$88$	−1.55638	−0.165911
$89$	9.21616	0.976911	0.488455	−	0.872589i	$-0.337561\pi$
$89$	9.21616	0.976911	0.488455	+	0.872589i	$0.337561\pi$
$90$	−9.21401	−0.971242
$91$	0.674303	0.0706861
$92$	4.34340	0.452831
$93$	12.9427	1.34210
$94$	−2.85379	−0.294346
$95$	−2.11518	−0.217013
$96$	−3.28301	−0.335070
$97$	6.15074	0.624513	0.312257	−	0.949998i	$-0.398915\pi$
$97$	6.15074	0.624513	0.312257	+	0.949998i	$0.398915\pi$
$98$	−6.90091	−0.697097
$99$	−12.1057	−1.21667
$100$	−3.59671	−0.359671
$101$	7.88310	0.784398	0.392199	−	0.919880i	$-0.371715\pi$
$101$	7.88310	0.784398	0.392199	+	0.919880i	$0.371715\pi$
$102$	11.5715	1.14575
$103$	4.01162	0.395276	0.197638	−	0.980275i	$-0.436673\pi$
$103$	4.01162	0.395276	0.197638	+	0.980275i	$0.436673\pi$
$104$	2.14208	0.210048
$105$	1.22424	0.119473
$106$	0.974012	0.0946044
$107$	16.5113	1.59621	0.798105	−	0.602518i	$-0.205836\pi$
$107$	16.5113	1.59621	0.798105	+	0.602518i	$0.205836\pi$
$108$	−15.6866	−1.50945
$109$	4.75246	0.455203	0.227601	−	0.973754i	$-0.426912\pi$
$109$	4.75246	0.455203	0.227601	+	0.973754i	$0.426912\pi$
$110$	1.84370	0.175789
$111$	−16.3808	−1.55479
$112$	0.314789	0.0297448
$113$	−5.99218	−0.563697	−0.281849	−	0.959459i	$-0.590948\pi$
$113$	−5.99218	−0.563697	−0.281849	+	0.959459i	$0.590948\pi$
$114$	−5.86199	−0.549025
$115$	−5.14522	−0.479794
$116$	−4.40062	−0.408587
$117$	16.6613	1.54034
$118$	3.56507	0.328191
$119$	−1.10952	−0.101710
$120$	3.88907	0.355022
$121$	−8.57768	−0.779790
$122$	10.6858	0.967447
$123$	−33.1606	−2.98999
$124$	−3.94235	−0.354033
$125$	10.1837	0.910859
$126$	2.44847	0.218127
$127$	−2.95356	−0.262086	−0.131043	−	0.991377i	$-0.541833\pi$
$127$	−2.95356	−0.262086	−0.131043	+	0.991377i	$0.541833\pi$
$128$	1.00000	0.0883883
$129$	23.6683	2.08388
$130$	−2.53752	−0.222555
$131$	13.9115	1.21546	0.607728	−	0.794145i	$-0.292081\pi$
$131$	13.9115	1.21546	0.607728	+	0.794145i	$0.292081\pi$
$132$	5.10960	0.444734
$133$	0.562074	0.0487380
$134$	2.19986	0.190039
$135$	18.5825	1.59932
$136$	−3.52465	−0.302237
$137$	−15.2268	−1.30091	−0.650457	−	0.759543i	$-0.725422\pi$
$137$	−15.2268	−1.30091	−0.650457	+	0.759543i	$0.725422\pi$
$138$	−14.2594	−1.21384
$139$	−21.6935	−1.84001	−0.920007	−	0.391901i	$-0.871818\pi$
$139$	−21.6935	−1.84001	−0.920007	+	0.391901i	$0.871818\pi$
$140$	−0.372901	−0.0315159
$141$	9.36900	0.789012
$142$	−14.5629	−1.22209
$143$	−3.33388	−0.278793
$144$	7.77813	0.648177
$145$	5.21299	0.432916
$146$	1.87294	0.155005
$147$	22.6557	1.86861
$148$	4.98956	0.410139
$149$	−14.9775	−1.22701	−0.613503	−	0.789692i	$-0.710240\pi$
$149$	−14.9775	−1.22701	−0.613503	+	0.789692i	$0.710240\pi$
$150$	11.8080	0.964121
$151$	−1.23674	−0.100645	−0.0503224	−	0.998733i	$-0.516025\pi$
$151$	−1.23674	−0.100645	−0.0503224	+	0.998733i	$0.516025\pi$
$152$	1.78556	0.144828
$153$	−27.4152	−2.21639
$154$	−0.489931	−0.0394798
$155$	4.67012	0.375113
$156$	−7.03245	−0.563047
$157$	−5.78774	−0.461912	−0.230956	−	0.972964i	$-0.574185\pi$
$157$	−5.78774	−0.461912	−0.230956	+	0.972964i	$0.574185\pi$
$158$	−5.95226	−0.473537
$159$	−3.19769	−0.253593
$160$	−1.18461	−0.0936513
$161$	1.36726	0.107755
$162$	28.1649	2.21284
$163$	11.4189	0.894394	0.447197	−	0.894435i	$-0.352422\pi$
$163$	11.4189	0.894394	0.447197	+	0.894435i	$0.352422\pi$
$164$	10.1007	0.788731
$165$	−6.05286	−0.471215
$166$	0.696817	0.0540835
$167$	8.36520	0.647319	0.323659	−	0.946174i	$-0.395087\pi$
$167$	8.36520	0.647319	0.323659	+	0.946174i	$0.395087\pi$
$168$	−1.03346	−0.0797328
$169$	−8.41151	−0.647039
$170$	4.17533	0.320233
$171$	13.8883	1.06206
$172$	−7.20934	−0.549707
$173$	18.1813	1.38230	0.691151	−	0.722711i	$-0.257104\pi$
$173$	18.1813	1.38230	0.691151	+	0.722711i	$0.257104\pi$
$174$	14.4472	1.09524
$175$	−1.13221	−0.0855867
$176$	−1.55638	−0.117316
$177$	−11.7041	−0.879737
$178$	9.21616	0.690780
$179$	−11.2628	−0.841821	−0.420910	−	0.907102i	$-0.638289\pi$
$179$	−11.2628	−0.841821	−0.420910	+	0.907102i	$0.638289\pi$
$180$	−9.21401	−0.686772
$181$	−22.9889	−1.70875	−0.854377	−	0.519654i	$-0.826061\pi$
$181$	−22.9889	−1.70875	−0.854377	+	0.519654i	$0.826061\pi$
$182$	0.674303	0.0499827
$183$	−35.0816	−2.59330
$184$	4.34340	0.320200
$185$	−5.91066	−0.434561
$186$	12.9427	0.949008
$187$	5.48570	0.401154
$188$	−2.85379	−0.208134
$189$	−4.93798	−0.359185
$190$	−2.11518	−0.153451
$191$	−4.53222	−0.327940	−0.163970	−	0.986465i	$-0.552430\pi$
$191$	−4.53222	−0.327940	−0.163970	+	0.986465i	$0.552430\pi$
$192$	−3.28301	−0.236931
$193$	−13.5505	−0.975389	−0.487694	−	0.873014i	$-0.662162\pi$
$193$	−13.5505	−0.975389	−0.487694	+	0.873014i	$0.662162\pi$
$194$	6.15074	0.441597
$195$	8.33068	0.596572
$196$	−6.90091	−0.492922
$197$	−5.03941	−0.359043	−0.179521	−	0.983754i	$-0.557455\pi$
$197$	−5.03941	−0.359043	−0.179521	+	0.983754i	$0.557455\pi$
$198$	−12.1057	−0.860316
$199$	18.2074	1.29069	0.645344	−	0.763892i	$-0.276714\pi$
$199$	18.2074	1.29069	0.645344	+	0.763892i	$0.276714\pi$
$200$	−3.59671	−0.254326
$201$	−7.22215	−0.509411
$202$	7.88310	0.554653
$203$	−1.38527	−0.0972267
$204$	11.5715	0.810164
$205$	−11.9653	−0.835695
$206$	4.01162	0.279503
$207$	33.7835	2.34812
$208$	2.14208	0.148526
$209$	−2.77900	−0.192227
$210$	1.22424	0.0844804
$211$	16.0298	1.10354	0.551769	−	0.833997i	$-0.313953\pi$
$211$	16.0298	1.10354	0.551769	+	0.833997i	$0.313953\pi$
$212$	0.974012	0.0668954
$213$	47.8100	3.27588
$214$	16.5113	1.12869
$215$	8.54023	0.582439
$216$	−15.6866	−1.06734
$217$	−1.24101	−0.0842451
$218$	4.75246	0.321877
$219$	−6.14886	−0.415501
$220$	1.84370	0.124302
$221$	−7.55008	−0.507873
$222$	−16.3808	−1.09940
$223$	−3.53961	−0.237030	−0.118515	−	0.992952i	$-0.537813\pi$
$223$	−3.53961	−0.237030	−0.118515	+	0.992952i	$0.537813\pi$
$224$	0.314789	0.0210327
$225$	−27.9757	−1.86504
$226$	−5.99218	−0.398594
$227$	−3.09990	−0.205748	−0.102874	−	0.994694i	$-0.532804\pi$
$227$	−3.09990	−0.205748	−0.102874	+	0.994694i	$0.532804\pi$
$228$	−5.86199	−0.388220
$229$	15.8890	1.04997	0.524987	−	0.851110i	$-0.324070\pi$
$229$	15.8890	1.04997	0.524987	+	0.851110i	$0.324070\pi$
$230$	−5.14522	−0.339266
$231$	1.60845	0.105828
$232$	−4.40062	−0.288915
$233$	5.12524	0.335766	0.167883	−	0.985807i	$-0.446307\pi$
$233$	5.12524	0.335766	0.167883	+	0.985807i	$0.446307\pi$
$234$	16.6613	1.08919
$235$	3.38061	0.220527
$236$	3.56507	0.232066
$237$	19.5413	1.26934
$238$	−1.10952	−0.0719197
$239$	−20.5253	−1.32767	−0.663836	−	0.747878i	$-0.731073\pi$
$239$	−20.5253	−1.32767	−0.663836	+	0.747878i	$0.731073\pi$
$240$	3.88907	0.251038
$241$	18.1055	1.16628	0.583140	−	0.812371i	$-0.301824\pi$
$241$	18.1055	1.16628	0.583140	+	0.812371i	$0.301824\pi$
$242$	−8.57768	−0.551394
$243$	−45.4056	−2.91277
$244$	10.6858	0.684088
$245$	8.17485	0.522272
$246$	−33.1606	−2.11424
$247$	3.82480	0.243366
$248$	−3.94235	−0.250339
$249$	−2.28766	−0.144974
$250$	10.1837	0.644074
$251$	−22.3580	−1.41123	−0.705614	−	0.708597i	$-0.749329\pi$
$251$	−22.3580	−1.41123	−0.705614	+	0.708597i	$0.749329\pi$
$252$	2.44847	0.154239
$253$	−6.75998	−0.424996
$254$	−2.95356	−0.185323
$255$	−13.7076	−0.858404
$256$	1.00000	0.0625000
$257$	−12.2519	−0.764256	−0.382128	−	0.924109i	$-0.624809\pi$
$257$	−12.2519	−0.764256	−0.382128	+	0.924109i	$0.624809\pi$
$258$	23.6683	1.47353
$259$	1.57066	0.0975961
$260$	−2.53752	−0.157370
$261$	−34.2286	−2.11869
$262$	13.9115	0.859457
$263$	−22.7350	−1.40190	−0.700950	−	0.713210i	$-0.747240\pi$
$263$	−22.7350	−1.40190	−0.700950	+	0.713210i	$0.747240\pi$
$264$	5.10960	0.314474
$265$	−1.15382	−0.0708786
$266$	0.562074	0.0344629
$267$	−30.2567	−1.85168
$268$	2.19986	0.134378
$269$	−31.2430	−1.90492	−0.952459	−	0.304666i	$-0.901455\pi$
$269$	−31.2430	−1.90492	−0.952459	+	0.304666i	$0.901455\pi$
$270$	18.5825	1.13089
$271$	−27.7414	−1.68517	−0.842586	−	0.538561i	$-0.818968\pi$
$271$	−27.7414	−1.68517	−0.842586	+	0.538561i	$0.818968\pi$
$272$	−3.52465	−0.213714
$273$	−2.21374	−0.133982
$274$	−15.2268	−0.919885
$275$	5.59784	0.337563
$276$	−14.2594	−0.858316
$277$	17.1261	1.02901	0.514505	−	0.857487i	$-0.327976\pi$
$277$	17.1261	1.02901	0.514505	+	0.857487i	$0.327976\pi$
$278$	−21.6935	−1.30109
$279$	−30.6641	−1.83581
$280$	−0.372901	−0.0222851
$281$	24.6927	1.47304	0.736521	−	0.676414i	$-0.236467\pi$
$281$	24.6927	1.47304	0.736521	+	0.676414i	$0.236467\pi$
$282$	9.36900	0.557916
$283$	−13.8961	−0.826035	−0.413017	−	0.910723i	$-0.635525\pi$
$283$	−13.8961	−0.826035	−0.413017	+	0.910723i	$0.635525\pi$
$284$	−14.5629	−0.864147
$285$	6.94414	0.411336
$286$	−3.33388	−0.197137
$287$	3.17959	0.187685
$288$	7.77813	0.458331
$289$	−4.57681	−0.269224
$290$	5.21299	0.306118
$291$	−20.1929	−1.18373
$292$	1.87294	0.109605
$293$	30.5705	1.78595	0.892973	−	0.450111i	$-0.148616\pi$
$293$	30.5705	1.78595	0.892973	+	0.450111i	$0.148616\pi$
$294$	22.6557	1.32131
$295$	−4.22320	−0.245884
$296$	4.98956	0.290012
$297$	24.4143	1.41666
$298$	−14.9775	−0.867624
$299$	9.30390	0.538058
$300$	11.8080	0.681736
$301$	−2.26942	−0.130807
$302$	−1.23674	−0.0711666
$303$	−25.8803	−1.48678
$304$	1.78556	0.102409
$305$	−12.6585	−0.724821
$306$	−27.4152	−1.56722
$307$	−8.82334	−0.503575	−0.251787	−	0.967783i	$-0.581018\pi$
$307$	−8.82334	−0.503575	−0.251787	+	0.967783i	$0.581018\pi$
$308$	−0.489931	−0.0279164
$309$	−13.1702	−0.749225
$310$	4.67012	0.265245
$311$	−25.1625	−1.42683	−0.713416	−	0.700741i	$-0.752853\pi$
$311$	−25.1625	−1.42683	−0.713416	+	0.700741i	$0.752853\pi$
$312$	−7.03245	−0.398134
$313$	−33.5302	−1.89524	−0.947620	−	0.319399i	$-0.896519\pi$
$313$	−33.5302	−1.89524	−0.947620	+	0.319399i	$0.896519\pi$
$314$	−5.78774	−0.326621
$315$	−2.90047	−0.163423
$316$	−5.95226	−0.334841
$317$	14.3096	0.803709	0.401855	−	0.915703i	$-0.368366\pi$
$317$	14.3096	0.803709	0.401855	+	0.915703i	$0.368366\pi$
$318$	−3.19769	−0.179317
$319$	6.84902	0.383472
$320$	−1.18461	−0.0662215
$321$	−54.2068	−3.02553
$322$	1.36726	0.0761942
$323$	−6.29346	−0.350178
$324$	28.1649	1.56472
$325$	−7.70443	−0.427365
$326$	11.4189	0.632432
$327$	−15.6023	−0.862811
$328$	10.1007	0.557717
$329$	−0.898342	−0.0495272
$330$	−6.05286	−0.333199
$331$	−22.9091	−1.25920	−0.629599	−	0.776920i	$-0.716781\pi$
$331$	−22.9091	−1.25920	−0.629599	+	0.776920i	$0.716781\pi$
$332$	0.696817	0.0382428
$333$	38.8095	2.12674
$334$	8.36520	0.457724
$335$	−2.60596	−0.142379
$336$	−1.03346	−0.0563796
$337$	−13.8111	−0.752337	−0.376168	−	0.926551i	$-0.622759\pi$
$337$	−13.8111	−0.752337	−0.376168	+	0.926551i	$0.622759\pi$
$338$	−8.41151	−0.457526
$339$	19.6724	1.06846
$340$	4.17533	0.226439
$341$	6.13578	0.332271
$342$	13.8883	0.750992
$343$	−4.37586	−0.236274
$344$	−7.20934	−0.388702
$345$	16.8918	0.909423
$346$	18.1813	0.977435
$347$	−16.8787	−0.906095	−0.453048	−	0.891486i	$-0.649663\pi$
$347$	−16.8787	−0.906095	−0.453048	+	0.891486i	$0.649663\pi$
$348$	14.4472	0.774454
$349$	4.63539	0.248127	0.124063	−	0.992274i	$-0.460407\pi$
$349$	4.63539	0.248127	0.124063	+	0.992274i	$0.460407\pi$
$350$	−1.13221	−0.0605189
$351$	−33.6019	−1.79354
$352$	−1.55638	−0.0829553
$353$	0.591176	0.0314651	0.0157326	−	0.999876i	$-0.494992\pi$
$353$	0.591176	0.0314651	0.0157326	+	0.999876i	$0.494992\pi$
$354$	−11.7041	−0.622068
$355$	17.2513	0.915601
$356$	9.21616	0.488455
$357$	3.64257	0.192785
$358$	−11.2628	−0.595257
$359$	9.15333	0.483094	0.241547	−	0.970389i	$-0.422345\pi$
$359$	9.15333	0.483094	0.241547	+	0.970389i	$0.422345\pi$
$360$	−9.21401	−0.485621
$361$	−15.8118	−0.832200
$362$	−22.9889	−1.20827
$363$	28.1606	1.47805
$364$	0.674303	0.0353431
$365$	−2.21869	−0.116132
$366$	−35.0816	−1.83374
$367$	−10.4258	−0.544221	−0.272111	−	0.962266i	$-0.587722\pi$
$367$	−10.4258	−0.544221	−0.272111	+	0.962266i	$0.587722\pi$
$368$	4.34340	0.226415
$369$	78.5645	4.08990
$370$	−5.91066	−0.307281
$371$	0.306609	0.0159183
$372$	12.9427	0.671050
$373$	−18.7669	−0.971715	−0.485858	−	0.874038i	$-0.661493\pi$
$373$	−18.7669	−0.971715	−0.485858	+	0.874038i	$0.661493\pi$
$374$	5.48570	0.283659
$375$	−33.4332	−1.72648
$376$	−2.85379	−0.147173
$377$	−9.42646	−0.485487
$378$	−4.93798	−0.253982
$379$	−19.5558	−1.00451	−0.502256	−	0.864719i	$-0.667497\pi$
$379$	−19.5558	−1.00451	−0.502256	+	0.864719i	$0.667497\pi$
$380$	−2.11518	−0.108506
$381$	9.69654	0.496769
$382$	−4.53222	−0.231889
$383$	−2.44330	−0.124847	−0.0624235	−	0.998050i	$-0.519883\pi$
$383$	−2.44330	−0.124847	−0.0624235	+	0.998050i	$0.519883\pi$
$384$	−3.28301	−0.167535
$385$	0.580376	0.0295787
$386$	−13.5505	−0.689704
$387$	−56.0752	−2.85046
$388$	6.15074	0.312257
$389$	31.8058	1.61262	0.806308	−	0.591496i	$-0.201462\pi$
$389$	31.8058	1.61262	0.806308	+	0.591496i	$0.201462\pi$
$390$	8.33068	0.421840
$391$	−15.3090	−0.774209
$392$	−6.90091	−0.348548
$393$	−45.6716	−2.30383
$394$	−5.03941	−0.253882
$395$	7.05108	0.354779
$396$	−12.1057	−0.608335
$397$	−24.1070	−1.20990	−0.604948	−	0.796265i	$-0.706806\pi$
$397$	−24.1070	−1.20990	−0.604948	+	0.796265i	$0.706806\pi$
$398$	18.2074	0.912655
$399$	−1.84529	−0.0923801
$400$	−3.59671	−0.179835
$401$	−9.14644	−0.456751	−0.228376	−	0.973573i	$-0.573341\pi$
$401$	−9.14644	−0.456751	−0.228376	+	0.973573i	$0.573341\pi$
$402$	−7.22215	−0.360208
$403$	−8.44481	−0.420666
$404$	7.88310	0.392199
$405$	−33.3643	−1.65788
$406$	−1.38527	−0.0687496
$407$	−7.76565	−0.384929
$408$	11.5715	0.572873
$409$	9.50038	0.469764	0.234882	−	0.972024i	$-0.424530\pi$
$409$	9.50038	0.469764	0.234882	+	0.972024i	$0.424530\pi$
$410$	−11.9653	−0.590926
$411$	49.9897	2.46581
$412$	4.01162	0.197638
$413$	1.12225	0.0552221
$414$	33.7835	1.66037
$415$	−0.825454	−0.0405199
$416$	2.14208	0.105024
$417$	71.2197	3.48765
$418$	−2.77900	−0.135925
$419$	−13.8658	−0.677389	−0.338694	−	0.940896i	$-0.609985\pi$
$419$	−13.8658	−0.677389	−0.338694	+	0.940896i	$0.609985\pi$
$420$	1.22424	0.0597367
$421$	−27.2413	−1.32766	−0.663829	−	0.747884i	$-0.731070\pi$
$421$	−27.2413	−1.32766	−0.663829	+	0.747884i	$0.731070\pi$
$422$	16.0298	0.780319
$423$	−22.1971	−1.07926
$424$	0.974012	0.0473022
$425$	12.6772	0.614932
$426$	47.8100	2.31640
$427$	3.36378	0.162785
$428$	16.5113	0.798105
$429$	10.9452	0.528437
$430$	8.54023	0.411847
$431$	−8.94877	−0.431047	−0.215523	−	0.976499i	$-0.569146\pi$
$431$	−8.94877	−0.431047	−0.215523	+	0.976499i	$0.569146\pi$
$432$	−15.6866	−0.754723
$433$	9.72491	0.467349	0.233675	−	0.972315i	$-0.424925\pi$
$433$	9.72491	0.467349	0.233675	+	0.972315i	$0.424925\pi$
$434$	−1.24101	−0.0595703
$435$	−17.1143	−0.820567
$436$	4.75246	0.227601
$437$	7.75538	0.370990
$438$	−6.14886	−0.293804
$439$	−19.5923	−0.935090	−0.467545	−	0.883969i	$-0.654861\pi$
$439$	−19.5923	−0.935090	−0.467545	+	0.883969i	$0.654861\pi$
$440$	1.84370	0.0878947
$441$	−53.6761	−2.55601
$442$	−7.55008	−0.359121
$443$	22.4462	1.06645	0.533224	−	0.845974i	$-0.320980\pi$
$443$	22.4462	1.06645	0.533224	+	0.845974i	$0.320980\pi$
$444$	−16.3808	−0.777397
$445$	−10.9175	−0.517540
$446$	−3.53961	−0.167605
$447$	49.1713	2.32572
$448$	0.314789	0.0148724
$449$	−5.01107	−0.236487	−0.118243	−	0.992985i	$-0.537726\pi$
$449$	−5.01107	−0.236487	−0.118243	+	0.992985i	$0.537726\pi$
$450$	−27.9757	−1.31879
$451$	−15.7205	−0.740249
$452$	−5.99218	−0.281849
$453$	4.06023	0.190766
$454$	−3.09990	−0.145486
$455$	−0.798783	−0.0374475
$456$	−5.86199	−0.274513
$457$	−30.5086	−1.42713	−0.713565	−	0.700589i	$-0.752921\pi$
$457$	−30.5086	−1.42713	−0.713565	+	0.700589i	$0.752921\pi$
$458$	15.8890	0.742443
$459$	55.2899	2.58071
$460$	−5.14522	−0.239897
$461$	29.2676	1.36313	0.681564	−	0.731758i	$-0.261300\pi$
$461$	29.2676	1.36313	0.681564	+	0.731758i	$0.261300\pi$
$462$	1.60845	0.0748318
$463$	−18.2701	−0.849085	−0.424543	−	0.905408i	$-0.639565\pi$
$463$	−18.2701	−0.849085	−0.424543	+	0.905408i	$0.639565\pi$
$464$	−4.40062	−0.204293
$465$	−15.3320	−0.711007
$466$	5.12524	0.237422
$467$	−5.50148	−0.254578	−0.127289	−	0.991866i	$-0.540628\pi$
$467$	−5.50148	−0.254578	−0.127289	+	0.991866i	$0.540628\pi$
$468$	16.6613	0.770171
$469$	0.692492	0.0319763
$470$	3.38061	0.155936
$471$	19.0012	0.875528
$472$	3.56507	0.164096
$473$	11.2205	0.515918
$474$	19.5413	0.897562
$475$	−6.42212	−0.294667
$476$	−1.10952	−0.0508549
$477$	7.57599	0.346881
$478$	−20.5253	−0.938806
$479$	9.09306	0.415472	0.207736	−	0.978185i	$-0.433390\pi$
$479$	9.09306	0.415472	0.207736	+	0.978185i	$0.433390\pi$
$480$	3.88907	0.177511
$481$	10.6880	0.487332
$482$	18.1055	0.824685
$483$	−4.48871	−0.204243
$484$	−8.57768	−0.389895
$485$	−7.28620	−0.330849
$486$	−45.4056	−2.05964
$487$	39.5730	1.79322	0.896612	−	0.442817i	$-0.146021\pi$
$487$	39.5730	1.79322	0.896612	+	0.442817i	$0.146021\pi$
$488$	10.6858	0.483724
$489$	−37.4882	−1.69527
$490$	8.17485	0.369302
$491$	−18.9570	−0.855519	−0.427760	−	0.903893i	$-0.640697\pi$
$491$	−18.9570	−0.855519	−0.427760	+	0.903893i	$0.640697\pi$
$492$	−33.1606	−1.49500
$493$	15.5106	0.698564
$494$	3.82480	0.172086
$495$	14.3405	0.644557
$496$	−3.94235	−0.177017
$497$	−4.58423	−0.205631
$498$	−2.28766	−0.102512
$499$	13.9566	0.624785	0.312393	−	0.949953i	$-0.398870\pi$
$499$	13.9566	0.624785	0.312393	+	0.949953i	$0.398870\pi$
$500$	10.1837	0.455429
$501$	−27.4630	−1.22696
$502$	−22.3580	−0.997888
$503$	3.75358	0.167364	0.0836820	−	0.996493i	$-0.473332\pi$
$503$	3.75358	0.167364	0.0836820	+	0.996493i	$0.473332\pi$
$504$	2.44847	0.109064
$505$	−9.33837	−0.415552
$506$	−6.75998	−0.300518
$507$	27.6150	1.22643
$508$	−2.95356	−0.131043
$509$	5.23507	0.232041	0.116020	−	0.993247i	$-0.462986\pi$
$509$	5.23507	0.232041	0.116020	+	0.993247i	$0.462986\pi$
$510$	−13.7076	−0.606984
$511$	0.589580	0.0260815
$512$	1.00000	0.0441942
$513$	−28.0093	−1.23664
$514$	−12.2519	−0.540410
$515$	−4.75219	−0.209406
$516$	23.6683	1.04194
$517$	4.44157	0.195340
$518$	1.57066	0.0690109
$519$	−59.6894	−2.62008
$520$	−2.53752	−0.111277
$521$	−17.5265	−0.767852	−0.383926	−	0.923364i	$-0.625428\pi$
$521$	−17.5265	−0.767852	−0.383926	+	0.923364i	$0.625428\pi$
$522$	−34.2286	−1.49814
$523$	−35.3791	−1.54702	−0.773509	−	0.633785i	$-0.781500\pi$
$523$	−35.3791	−1.54702	−0.773509	+	0.633785i	$0.781500\pi$
$524$	13.9115	0.607728
$525$	3.71704	0.162225
$526$	−22.7350	−0.991293
$527$	13.8954	0.605293
$528$	5.10960	0.222367
$529$	−4.13486	−0.179777
$530$	−1.15382	−0.0501188
$531$	27.7296	1.20336
$532$	0.562074	0.0243690
$533$	21.6364	0.937179
$534$	−30.2567	−1.30934
$535$	−19.5594	−0.845627
$536$	2.19986	0.0950194
$537$	36.9758	1.59562
$538$	−31.2430	−1.34698
$539$	10.7404	0.462623
$540$	18.5825	0.799662
$541$	−32.3828	−1.39225	−0.696123	−	0.717922i	$-0.745093\pi$
$541$	−32.3828	−1.39225	−0.696123	+	0.717922i	$0.745093\pi$
$542$	−27.7414	−1.19160
$543$	75.4728	3.23885
$544$	−3.52465	−0.151118
$545$	−5.62979	−0.241154
$546$	−2.21374	−0.0947393
$547$	16.0202	0.684972	0.342486	−	0.939523i	$-0.388731\pi$
$547$	16.0202	0.684972	0.342486	+	0.939523i	$0.388731\pi$
$548$	−15.2268	−0.650457
$549$	83.1155	3.54728
$550$	5.59784	0.238693
$551$	−7.85754	−0.334743
$552$	−14.2594	−0.606921
$553$	−1.87371	−0.0796782
$554$	17.1261	0.727620
$555$	19.4047	0.823685
$556$	−21.6935	−0.920007
$557$	3.91864	0.166038	0.0830191	−	0.996548i	$-0.473544\pi$
$557$	3.91864	0.166038	0.0830191	+	0.996548i	$0.473544\pi$
$558$	−30.6641	−1.29811
$559$	−15.4430	−0.653168
$560$	−0.372901	−0.0157580
$561$	−18.0096	−0.760365
$562$	24.6927	1.04160
$563$	13.0201	0.548733	0.274367	−	0.961625i	$-0.411532\pi$
$563$	13.0201	0.548733	0.274367	+	0.961625i	$0.411532\pi$
$564$	9.36900	0.394506
$565$	7.09838	0.298631
$566$	−13.8961	−0.584095
$567$	8.86601	0.372337
$568$	−14.5629	−0.611044
$569$	4.48677	0.188095	0.0940475	−	0.995568i	$-0.470019\pi$
$569$	4.48677	0.188095	0.0940475	+	0.995568i	$0.470019\pi$
$570$	6.94414	0.290858
$571$	−28.6479	−1.19888	−0.599439	−	0.800420i	$-0.704610\pi$
$571$	−28.6479	−1.19888	−0.599439	+	0.800420i	$0.704610\pi$
$572$	−3.33388	−0.139397
$573$	14.8793	0.621592
$574$	3.17959	0.132713
$575$	−15.6220	−0.651480
$576$	7.77813	0.324089
$577$	31.0574	1.29294	0.646468	−	0.762941i	$-0.276245\pi$
$577$	31.0574	1.29294	0.646468	+	0.762941i	$0.276245\pi$
$578$	−4.57681	−0.190370
$579$	44.4865	1.84879
$580$	5.21299	0.216458
$581$	0.219351	0.00910020
$582$	−20.1929	−0.837023
$583$	−1.51593	−0.0627835
$584$	1.87294	0.0775026
$585$	−19.7371	−0.816030
$586$	30.5705	1.26285
$587$	−3.46927	−0.143192	−0.0715960	−	0.997434i	$-0.522809\pi$
$587$	−3.46927	−0.143192	−0.0715960	+	0.997434i	$0.522809\pi$
$588$	22.6557	0.934306
$589$	−7.03927	−0.290048
$590$	−4.22320	−0.173866
$591$	16.5444	0.680546
$592$	4.98956	0.205070
$593$	32.8666	1.34967	0.674835	−	0.737969i	$-0.264215\pi$
$593$	32.8666	1.34967	0.674835	+	0.737969i	$0.264215\pi$
$594$	24.4143	1.00173
$595$	1.31435	0.0538830
$596$	−14.9775	−0.613503
$597$	−59.7750	−2.44643
$598$	9.30390	0.380465
$599$	−14.8545	−0.606940	−0.303470	−	0.952841i	$-0.598145\pi$
$599$	−14.8545	−0.606940	−0.303470	+	0.952841i	$0.598145\pi$
$600$	11.8080	0.482060
$601$	−31.9831	−1.30462	−0.652310	−	0.757953i	$-0.726200\pi$
$601$	−31.9831	−1.30462	−0.652310	+	0.757953i	$0.726200\pi$
$602$	−2.26942	−0.0924949
$603$	17.1108	0.696805
$604$	−1.23674	−0.0503224
$605$	10.1612	0.413110
$606$	−25.8803	−1.05131
$607$	−22.7883	−0.924947	−0.462473	−	0.886633i	$-0.653038\pi$
$607$	−22.7883	−0.924947	−0.462473	+	0.886633i	$0.653038\pi$
$608$	1.78556	0.0724138
$609$	4.54784	0.184288
$610$	−12.6585	−0.512526
$611$	−6.11303	−0.247307
$612$	−27.4152	−1.10819
$613$	34.7574	1.40384	0.701919	−	0.712257i	$-0.252327\pi$
$613$	34.7574	1.40384	0.701919	+	0.712257i	$0.252327\pi$
$614$	−8.82334	−0.356081
$615$	39.2823	1.58401
$616$	−0.489931	−0.0197399
$617$	19.7565	0.795367	0.397684	−	0.917523i	$-0.369814\pi$
$617$	19.7565	0.795367	0.397684	+	0.917523i	$0.369814\pi$
$618$	−13.1702	−0.529782
$619$	−1.31251	−0.0527541	−0.0263771	−	0.999652i	$-0.508397\pi$
$619$	−1.31251	−0.0527541	−0.0263771	+	0.999652i	$0.508397\pi$
$620$	4.67012	0.187557
$621$	−68.1333	−2.73410
$622$	−25.1625	−1.00892
$623$	2.90115	0.116232
$624$	−7.03245	−0.281523
$625$	5.91987	0.236795
$626$	−33.5302	−1.34014
$627$	9.12347	0.364356
$628$	−5.78774	−0.230956
$629$	−17.5865	−0.701219
$630$	−2.90047	−0.115558
$631$	−24.4591	−0.973703	−0.486851	−	0.873485i	$-0.661855\pi$
$631$	−24.4591	−0.973703	−0.486851	+	0.873485i	$0.661855\pi$
$632$	−5.95226	−0.236768
$633$	−52.6260	−2.09169
$634$	14.3096	0.568308
$635$	3.49880	0.138846
$636$	−3.19769	−0.126797
$637$	−14.7823	−0.585695
$638$	6.84902	0.271155
$639$	−113.272	−4.48096
$640$	−1.18461	−0.0468257
$641$	28.3237	1.11872	0.559360	−	0.828925i	$-0.311047\pi$
$641$	28.3237	1.11872	0.559360	+	0.828925i	$0.311047\pi$
$642$	−54.2068	−2.13937
$643$	27.3780	1.07968	0.539841	−	0.841767i	$-0.318485\pi$
$643$	27.3780	1.07968	0.539841	+	0.841767i	$0.318485\pi$
$644$	1.36726	0.0538775
$645$	−28.0376	−1.10398
$646$	−6.29346	−0.247613
$647$	−46.9494	−1.84577	−0.922886	−	0.385074i	$-0.874176\pi$
$647$	−46.9494	−1.84577	−0.922886	+	0.385074i	$0.874176\pi$
$648$	28.1649	1.10642
$649$	−5.54860	−0.217802
$650$	−7.70443	−0.302192
$651$	4.07424	0.159682
$652$	11.4189	0.447197
$653$	−13.2121	−0.517030	−0.258515	−	0.966007i	$-0.583233\pi$
$653$	−13.2121	−0.517030	−0.258515	+	0.966007i	$0.583233\pi$
$654$	−15.6023	−0.610100
$655$	−16.4797	−0.643914
$656$	10.1007	0.394366
$657$	14.5679	0.568349
$658$	−0.898342	−0.0350210
$659$	−5.84163	−0.227557	−0.113779	−	0.993506i	$-0.536295\pi$
$659$	−5.84163	−0.227557	−0.113779	+	0.993506i	$0.536295\pi$
$660$	−6.05286	−0.235607
$661$	1.74480	0.0678647	0.0339324	−	0.999424i	$-0.489197\pi$
$661$	1.74480	0.0678647	0.0339324	+	0.999424i	$0.489197\pi$
$662$	−22.9091	−0.890387
$663$	24.7870	0.962646
$664$	0.696817	0.0270418
$665$	−0.665836	−0.0258200
$666$	38.8095	1.50384
$667$	−19.1136	−0.740083
$668$	8.36520	0.323659
$669$	11.6206	0.449276
$670$	−2.60596	−0.100677
$671$	−16.6312	−0.642039
$672$	−1.03346	−0.0398664
$673$	12.0087	0.462900	0.231450	−	0.972847i	$-0.425653\pi$
$673$	12.0087	0.462900	0.231450	+	0.972847i	$0.425653\pi$
$674$	−13.8111	−0.531982
$675$	56.4202	2.17162
$676$	−8.41151	−0.323520
$677$	−1.05550	−0.0405660	−0.0202830	−	0.999794i	$-0.506457\pi$
$677$	−1.05550	−0.0405660	−0.0202830	+	0.999794i	$0.506457\pi$
$678$	19.6724	0.755513
$679$	1.93619	0.0743041
$680$	4.17533	0.160116
$681$	10.1770	0.389983
$682$	6.13578	0.234951
$683$	−46.6564	−1.78526	−0.892629	−	0.450792i	$-0.851142\pi$
$683$	−46.6564	−1.78526	−0.892629	+	0.450792i	$0.851142\pi$
$684$	13.8883	0.531031
$685$	18.0378	0.689187
$686$	−4.37586	−0.167071
$687$	−52.1636	−1.99017
$688$	−7.20934	−0.274854
$689$	2.08641	0.0794858
$690$	16.8918	0.643059
$691$	40.2089	1.52962	0.764810	−	0.644256i	$-0.222833\pi$
$691$	40.2089	1.52962	0.764810	+	0.644256i	$0.222833\pi$
$692$	18.1813	0.691151
$693$	−3.81075	−0.144758
$694$	−16.8787	−0.640706
$695$	25.6982	0.974788
$696$	14.4472	0.547621
$697$	−35.6014	−1.34850
$698$	4.63539	0.175452
$699$	−16.8262	−0.636426
$700$	−1.13221	−0.0427934
$701$	−36.4145	−1.37536	−0.687678	−	0.726016i	$-0.741370\pi$
$701$	−36.4145	−1.37536	−0.687678	+	0.726016i	$0.741370\pi$
$702$	−33.6019	−1.26822
$703$	8.90914	0.336014
$704$	−1.55638	−0.0586582
$705$	−11.0986	−0.417996
$706$	0.591176	0.0222492
$707$	2.48152	0.0933270
$708$	−11.7041	−0.439869
$709$	−2.80787	−0.105452	−0.0527258	−	0.998609i	$-0.516791\pi$
$709$	−2.80787	−0.105452	−0.0527258	+	0.998609i	$0.516791\pi$
$710$	17.2513	0.647428
$711$	−46.2975	−1.73629
$712$	9.21616	0.345390
$713$	−17.1232	−0.641268
$714$	3.64257	0.136320
$715$	3.94934	0.147697
$716$	−11.2628	−0.420910
$717$	67.3847	2.51653
$718$	9.15333	0.341599
$719$	−22.8609	−0.852567	−0.426283	−	0.904590i	$-0.640177\pi$
$719$	−22.8609	−0.852567	−0.426283	+	0.904590i	$0.640177\pi$
$720$	−9.21401	−0.343386
$721$	1.26281	0.0470297
$722$	−15.8118	−0.588454
$723$	−59.4406	−2.21062
$724$	−22.9889	−0.854377
$725$	15.8277	0.587827
$726$	28.1606	1.04514
$727$	−37.3778	−1.38627	−0.693133	−	0.720810i	$-0.743770\pi$
$727$	−37.3778	−1.38627	−0.693133	+	0.720810i	$0.743770\pi$
$728$	0.674303	0.0249913
$729$	64.5723	2.39157
$730$	−2.21869	−0.0821174
$731$	25.4104	0.939839
$732$	−35.0816	−1.29665
$733$	39.8983	1.47368	0.736839	−	0.676069i	$-0.236318\pi$
$733$	39.8983	1.47368	0.736839	+	0.676069i	$0.236318\pi$
$734$	−10.4258	−0.384822
$735$	−26.8381	−0.989938
$736$	4.34340	0.160100
$737$	−3.42381	−0.126118
$738$	78.5645	2.89200
$739$	−0.836021	−0.0307535	−0.0153768	−	0.999882i	$-0.504895\pi$
$739$	−0.836021	−0.0307535	−0.0153768	+	0.999882i	$0.504895\pi$
$740$	−5.91066	−0.217280
$741$	−12.5568	−0.461287
$742$	0.306609	0.0112560
$743$	−23.0838	−0.846862	−0.423431	−	0.905928i	$-0.639174\pi$
$743$	−23.0838	−0.846862	−0.423431	+	0.905928i	$0.639174\pi$
$744$	12.9427	0.474504
$745$	17.7424	0.650033
$746$	−18.7669	−0.687106
$747$	5.41994	0.198305
$748$	5.48570	0.200577
$749$	5.19759	0.189916
$750$	−33.4332	−1.22081
$751$	−21.3692	−0.779774	−0.389887	−	0.920863i	$-0.627486\pi$
$751$	−21.3692	−0.779774	−0.389887	+	0.920863i	$0.627486\pi$
$752$	−2.85379	−0.104067
$753$	73.4016	2.67490
$754$	−9.42646	−0.343291
$755$	1.46505	0.0533187
$756$	−4.93798	−0.179593
$757$	−51.3201	−1.86526	−0.932630	−	0.360834i	$-0.882492\pi$
$757$	−51.3201	−1.86526	−0.932630	+	0.360834i	$0.882492\pi$
$758$	−19.5558	−0.710297
$759$	22.1930	0.805557
$760$	−2.11518	−0.0767256
$761$	19.5193	0.707575	0.353787	−	0.935326i	$-0.384894\pi$
$761$	19.5193	0.707575	0.353787	+	0.935326i	$0.384894\pi$
$762$	9.69654	0.351269
$763$	1.49602	0.0541596
$764$	−4.53222	−0.163970
$765$	32.4762	1.17418
$766$	−2.44330	−0.0882801
$767$	7.63665	0.275743
$768$	−3.28301	−0.118465
$769$	28.0701	1.01223	0.506116	−	0.862465i	$-0.331081\pi$
$769$	28.0701	1.01223	0.506116	+	0.862465i	$0.331081\pi$
$770$	0.580376	0.0209153
$771$	40.2232	1.44860
$772$	−13.5505	−0.487694
$773$	6.77340	0.243622	0.121811	−	0.992553i	$-0.461130\pi$
$773$	6.77340	0.243622	0.121811	+	0.992553i	$0.461130\pi$
$774$	−56.0752	−2.01558
$775$	14.1795	0.509342
$776$	6.15074	0.220799
$777$	−5.15649	−0.184988
$778$	31.8058	1.14029
$779$	18.0353	0.646183
$780$	8.33068	0.298286
$781$	22.6653	0.811029
$782$	−15.3090	−0.547448
$783$	69.0308	2.46696
$784$	−6.90091	−0.246461
$785$	6.85619	0.244708
$786$	−45.6716	−1.62905
$787$	−42.5419	−1.51646	−0.758228	−	0.651989i	$-0.773935\pi$
$787$	−42.5419	−1.51646	−0.758228	+	0.651989i	$0.773935\pi$
$788$	−5.03941	−0.179521
$789$	74.6391	2.65722
$790$	7.05108	0.250866
$791$	−1.88628	−0.0670683
$792$	−12.1057	−0.430158
$793$	22.8898	0.812841
$794$	−24.1070	−0.855525
$795$	3.78800	0.134347
$796$	18.2074	0.645344
$797$	32.9122	1.16581	0.582906	−	0.812540i	$-0.301916\pi$
$797$	32.9122	1.16581	0.582906	+	0.812540i	$0.301916\pi$
$798$	−1.84529	−0.0653226
$799$	10.0586	0.355848
$800$	−3.59671	−0.127163
$801$	71.6845	2.53285
$802$	−9.14644	−0.322972
$803$	−2.91500	−0.102868
$804$	−7.22215	−0.254706
$805$	−1.61966	−0.0570855
$806$	−8.44481	−0.297456
$807$	102.571	3.61067
$808$	7.88310	0.277326
$809$	−12.0737	−0.424490	−0.212245	−	0.977216i	$-0.568078\pi$
$809$	−12.0737	−0.424490	−0.212245	+	0.977216i	$0.568078\pi$
$810$	−33.3643	−1.17230
$811$	2.60486	0.0914689	0.0457344	−	0.998954i	$-0.485437\pi$
$811$	2.60486	0.0914689	0.0457344	+	0.998954i	$0.485437\pi$
$812$	−1.38527	−0.0486133
$813$	91.0753	3.19415
$814$	−7.76565	−0.272186
$815$	−13.5268	−0.473825
$816$	11.5715	0.405082
$817$	−12.8727	−0.450358
$818$	9.50038	0.332173
$819$	5.24481	0.183269
$820$	−11.9653	−0.417848
$821$	−44.3430	−1.54758	−0.773791	−	0.633441i	$-0.781642\pi$
$821$	−44.3430	−1.54758	−0.773791	+	0.633441i	$0.781642\pi$
$822$	49.9897	1.74359
$823$	−0.476430	−0.0166073	−0.00830366	−	0.999966i	$-0.502643\pi$
$823$	−0.476430	−0.0166073	−0.00830366	+	0.999966i	$0.502643\pi$
$824$	4.01162	0.139751
$825$	−18.3777	−0.639831
$826$	1.12225	0.0390479
$827$	36.7087	1.27649	0.638243	−	0.769835i	$-0.279662\pi$
$827$	36.7087	1.27649	0.638243	+	0.769835i	$0.279662\pi$
$828$	33.7835	1.17406
$829$	−9.80510	−0.340545	−0.170273	−	0.985397i	$-0.554465\pi$
$829$	−9.80510	−0.340545	−0.170273	+	0.985397i	$0.554465\pi$
$830$	−0.825454	−0.0286519
$831$	−56.2252	−1.95043
$832$	2.14208	0.0742631
$833$	24.3233	0.842753
$834$	71.2197	2.46614
$835$	−9.90947	−0.342931
$836$	−2.77900	−0.0961137
$837$	61.8421	2.13758
$838$	−13.8658	−0.478986
$839$	−21.1446	−0.729991	−0.364996	−	0.931009i	$-0.618930\pi$
$839$	−21.1446	−0.729991	−0.364996	+	0.931009i	$0.618930\pi$
$840$	1.22424	0.0422402
$841$	−9.63458	−0.332227
$842$	−27.2413	−0.938796
$843$	−81.0663	−2.79207
$844$	16.0298	0.551769
$845$	9.96432	0.342783
$846$	−22.1971	−0.763153
$847$	−2.70016	−0.0927787
$848$	0.974012	0.0334477
$849$	45.6208	1.56570
$850$	12.6772	0.434823
$851$	21.6717	0.742895
$852$	47.8100	1.63794
$853$	25.2979	0.866183	0.433092	−	0.901350i	$-0.357423\pi$
$853$	25.2979	0.866183	0.433092	+	0.901350i	$0.357423\pi$
$854$	3.36378	0.115106
$855$	−16.4521	−0.562651
$856$	16.5113	0.564345
$857$	22.5508	0.770322	0.385161	−	0.922849i	$-0.374146\pi$
$857$	22.5508	0.770322	0.385161	+	0.922849i	$0.374146\pi$
$858$	10.9452	0.373661
$859$	41.3429	1.41060	0.705301	−	0.708908i	$-0.250812\pi$
$859$	41.3429	1.41060	0.705301	+	0.708908i	$0.250812\pi$
$860$	8.54023	0.291219
$861$	−10.4386	−0.355747
$862$	−8.94877	−0.304796
$863$	−10.2671	−0.349497	−0.174749	−	0.984613i	$-0.555911\pi$
$863$	−10.2671	−0.349497	−0.174749	+	0.984613i	$0.555911\pi$
$864$	−15.6866	−0.533670
$865$	−21.5377	−0.732304
$866$	9.72491	0.330466
$867$	15.0257	0.510300
$868$	−1.24101	−0.0421226
$869$	9.26398	0.314259
$870$	−17.1143	−0.580229
$871$	4.71227	0.159669
$872$	4.75246	0.160938
$873$	47.8412	1.61918
$874$	7.75538	0.262330
$875$	3.20572	0.108373
$876$	−6.14886	−0.207751
$877$	14.1010	0.476155	0.238078	−	0.971246i	$-0.423483\pi$
$877$	14.1010	0.476155	0.238078	+	0.971246i	$0.423483\pi$
$878$	−19.5923	−0.661208
$879$	−100.363	−3.38516
$880$	1.84370	0.0621510
$881$	−8.46166	−0.285081	−0.142540	−	0.989789i	$-0.545527\pi$
$881$	−8.46166	−0.285081	−0.142540	+	0.989789i	$0.545527\pi$
$882$	−53.6761	−1.80737
$883$	11.6796	0.393050	0.196525	−	0.980499i	$-0.437034\pi$
$883$	11.6796	0.393050	0.196525	+	0.980499i	$0.437034\pi$
$884$	−7.55008	−0.253937
$885$	13.8648	0.466060
$886$	22.4462	0.754093
$887$	−11.4267	−0.383671	−0.191836	−	0.981427i	$-0.561444\pi$
$887$	−11.4267	−0.383671	−0.191836	+	0.981427i	$0.561444\pi$
$888$	−16.3808	−0.549702
$889$	−0.929748	−0.0311827
$890$	−10.9175	−0.365956
$891$	−43.8352	−1.46854
$892$	−3.53961	−0.118515
$893$	−5.09559	−0.170518
$894$	49.1713	1.64453
$895$	13.3420	0.445973
$896$	0.314789	0.0105164
$897$	−30.5448	−1.01986
$898$	−5.01107	−0.167221
$899$	17.3487	0.578613
$900$	−27.9757	−0.932522
$901$	−3.43306	−0.114372
$902$	−15.7205	−0.523435
$903$	7.45054	0.247938
$904$	−5.99218	−0.199297
$905$	27.2328	0.905250
$906$	4.06023	0.134892
$907$	−51.0124	−1.69384	−0.846919	−	0.531722i	$-0.821545\pi$
$907$	−51.0124	−1.69384	−0.846919	+	0.531722i	$0.821545\pi$
$908$	−3.09990	−0.102874
$909$	61.3158	2.03372
$910$	−0.798783	−0.0264794
$911$	33.4166	1.10714	0.553571	−	0.832802i	$-0.313265\pi$
$911$	33.4166	1.10714	0.553571	+	0.832802i	$0.313265\pi$
$912$	−5.86199	−0.194110
$913$	−1.08451	−0.0358921
$914$	−30.5086	−1.00913
$915$	41.5578	1.37386
$916$	15.8890	0.524987
$917$	4.37920	0.144614
$918$	55.2899	1.82484
$919$	17.6039	0.580697	0.290349	−	0.956921i	$-0.406229\pi$
$919$	17.6039	0.580697	0.290349	+	0.956921i	$0.406229\pi$
$920$	−5.14522	−0.169633
$921$	28.9671	0.954497
$922$	29.2676	0.963877
$923$	−31.1948	−1.02679
$924$	1.60845	0.0529141
$925$	−17.9460	−0.590061
$926$	−18.2701	−0.600394
$927$	31.2029	1.02484
$928$	−4.40062	−0.144457
$929$	15.3177	0.502558	0.251279	−	0.967915i	$-0.419149\pi$
$929$	15.3177	0.502558	0.251279	+	0.967915i	$0.419149\pi$
$930$	−15.3320	−0.502758
$931$	−12.3220	−0.403836
$932$	5.12524	0.167883
$933$	82.6085	2.70448
$934$	−5.50148	−0.180014
$935$	−6.49839	−0.212520
$936$	16.6613	0.544593
$937$	31.4165	1.02633	0.513166	−	0.858289i	$-0.328473\pi$
$937$	31.4165	1.02633	0.513166	+	0.858289i	$0.328473\pi$
$938$	0.692492	0.0226107
$939$	110.080	3.59232
$940$	3.38061	0.110263
$941$	22.6058	0.736927	0.368464	−	0.929642i	$-0.379884\pi$
$941$	22.6058	0.736927	0.368464	+	0.929642i	$0.379884\pi$
$942$	19.0012	0.619092
$943$	43.8713	1.42865
$944$	3.56507	0.116033
$945$	5.84956	0.190286
$946$	11.2205	0.364809
$947$	34.4053	1.11802	0.559012	−	0.829160i	$-0.311181\pi$
$947$	34.4053	1.11802	0.559012	+	0.829160i	$0.311181\pi$
$948$	19.5413	0.634672
$949$	4.01197	0.130234
$950$	−6.42212	−0.208361
$951$	−46.9786	−1.52339
$952$	−1.10952	−0.0359599
$953$	40.8718	1.32397	0.661984	−	0.749518i	$-0.269715\pi$
$953$	40.8718	1.32397	0.661984	+	0.749518i	$0.269715\pi$
$954$	7.57599	0.245282
$955$	5.36889	0.173733
$956$	−20.5253	−0.663836
$957$	−22.4854	−0.726849
$958$	9.09306	0.293783
$959$	−4.79323	−0.154782
$960$	3.88907	0.125519
$961$	−15.4579	−0.498642
$962$	10.6880	0.344596
$963$	128.427	4.13851
$964$	18.1055	0.583140
$965$	16.0520	0.516733
$966$	−4.48871	−0.144422
$967$	−51.7717	−1.66487	−0.832433	−	0.554125i	$-0.813053\pi$
$967$	−51.7717	−1.66487	−0.832433	+	0.554125i	$0.813053\pi$
$968$	−8.57768	−0.275697
$969$	20.6615	0.663742
$970$	−7.28620	−0.233946
$971$	2.27440	0.0729889	0.0364945	−	0.999334i	$-0.488381\pi$
$971$	2.27440	0.0729889	0.0364945	+	0.999334i	$0.488381\pi$
$972$	−45.4056	−1.45639
$973$	−6.82887	−0.218923
$974$	39.5730	1.26800
$975$	25.2937	0.810046
$976$	10.6858	0.342044
$977$	18.8913	0.604385	0.302193	−	0.953247i	$-0.402281\pi$
$977$	18.8913	0.604385	0.302193	+	0.953247i	$0.402281\pi$
$978$	−37.4882	−1.19874
$979$	−14.3438	−0.458431
$980$	8.17485	0.261136
$981$	36.9652	1.18021
$982$	−18.9570	−0.604944
$983$	3.42239	0.109157	0.0545786	−	0.998509i	$-0.482618\pi$
$983$	3.42239	0.109157	0.0545786	+	0.998509i	$0.482618\pi$
$984$	−33.1606	−1.05712
$985$	5.96971	0.190211
$986$	15.5106	0.493960
$987$	2.94926	0.0938760
$988$	3.82480	0.121683
$989$	−31.3131	−0.995698
$990$	14.3405	0.455771
$991$	25.4032	0.806960	0.403480	−	0.914988i	$-0.367801\pi$
$991$	25.4032	0.806960	0.403480	+	0.914988i	$0.367801\pi$
$992$	−3.94235	−0.125170
$993$	75.2107	2.38674
$994$	−4.58423	−0.145403
$995$	−21.5686	−0.683771
$996$	−2.28766	−0.0724872
$997$	−30.1766	−0.955701	−0.477851	−	0.878441i	$-0.658584\pi$
$997$	−30.1766	−0.955701	−0.477851	+	0.878441i	$0.658584\pi$
$998$	13.9566	0.441790
$999$	−78.2694	−2.47633

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8006.2.a.a.1.2	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8006.2.a.a.1.2	✓	69	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8006 = 2 \cdot 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.28301 q^{3} +1.00000 q^{4} -1.18461 q^{5} -3.28301 q^{6} +0.314789 q^{7} +1.00000 q^{8} +7.77813 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.28301 q^{3} +1.00000 q^{4} -1.18461 q^{5} -3.28301 q^{6} +0.314789 q^{7} +1.00000 q^{8} +7.77813 q^{9} -1.18461 q^{10} -1.55638 q^{11} -3.28301 q^{12} +2.14208 q^{13} +0.314789 q^{14} +3.88907 q^{15} +1.00000 q^{16} -3.52465 q^{17} +7.77813 q^{18} +1.78556 q^{19} -1.18461 q^{20} -1.03346 q^{21} -1.55638 q^{22} +4.34340 q^{23} -3.28301 q^{24} -3.59671 q^{25} +2.14208 q^{26} -15.6866 q^{27} +0.314789 q^{28} -4.40062 q^{29} +3.88907 q^{30} -3.94235 q^{31} +1.00000 q^{32} +5.10960 q^{33} -3.52465 q^{34} -0.372901 q^{35} +7.77813 q^{36} +4.98956 q^{37} +1.78556 q^{38} -7.03245 q^{39} -1.18461 q^{40} +10.1007 q^{41} -1.03346 q^{42} -7.20934 q^{43} -1.55638 q^{44} -9.21401 q^{45} +4.34340 q^{46} -2.85379 q^{47} -3.28301 q^{48} -6.90091 q^{49} -3.59671 q^{50} +11.5715 q^{51} +2.14208 q^{52} +0.974012 q^{53} -15.6866 q^{54} +1.84370 q^{55} +0.314789 q^{56} -5.86199 q^{57} -4.40062 q^{58} +3.56507 q^{59} +3.88907 q^{60} +10.6858 q^{61} -3.94235 q^{62} +2.44847 q^{63} +1.00000 q^{64} -2.53752 q^{65} +5.10960 q^{66} +2.19986 q^{67} -3.52465 q^{68} -14.2594 q^{69} -0.372901 q^{70} -14.5629 q^{71} +7.77813 q^{72} +1.87294 q^{73} +4.98956 q^{74} +11.8080 q^{75} +1.78556 q^{76} -0.489931 q^{77} -7.03245 q^{78} -5.95226 q^{79} -1.18461 q^{80} +28.1649 q^{81} +10.1007 q^{82} +0.696817 q^{83} -1.03346 q^{84} +4.17533 q^{85} -7.20934 q^{86} +14.4472 q^{87} -1.55638 q^{88} +9.21616 q^{89} -9.21401 q^{90} +0.674303 q^{91} +4.34340 q^{92} +12.9427 q^{93} -2.85379 q^{94} -2.11518 q^{95} -3.28301 q^{96} +6.15074 q^{97} -6.90091 q^{98} -12.1057 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9} - 9 q^{10} - 48 q^{11} - 15 q^{12} - 30 q^{13} - 29 q^{14} - 51 q^{15} + 69 q^{16} - 37 q^{17} + 40 q^{18} - 72 q^{19} - 9 q^{20} - 38 q^{21} - 48 q^{22} - 75 q^{23} - 15 q^{24} + 18 q^{25} - 30 q^{26} - 48 q^{27} - 29 q^{28} - 27 q^{29} - 51 q^{30} - 61 q^{31} + 69 q^{32} - 29 q^{33} - 37 q^{34} - 64 q^{35} + 40 q^{36} - 42 q^{37} - 72 q^{38} - 68 q^{39} - 9 q^{40} - 49 q^{41} - 38 q^{42} - 95 q^{43} - 48 q^{44} - 20 q^{45} - 75 q^{46} - 62 q^{47} - 15 q^{48} - 4 q^{49} + 18 q^{50} - 76 q^{51} - 30 q^{52} - 28 q^{53} - 48 q^{54} - 76 q^{55} - 29 q^{56} - 44 q^{57} - 27 q^{58} - 68 q^{59} - 51 q^{60} - 62 q^{61} - 61 q^{62} - 91 q^{63} + 69 q^{64} - 79 q^{65} - 29 q^{66} - 116 q^{67} - 37 q^{68} - 23 q^{69} - 64 q^{70} - 89 q^{71} + 40 q^{72} - 60 q^{73} - 42 q^{74} - 47 q^{75} - 72 q^{76} + 5 q^{77} - 68 q^{78} - 170 q^{79} - 9 q^{80} - 3 q^{81} - 49 q^{82} - 82 q^{83} - 38 q^{84} - 81 q^{85} - 95 q^{86} - 51 q^{87} - 48 q^{88} - 78 q^{89} - 20 q^{90} - 85 q^{91} - 75 q^{92} - 21 q^{93} - 62 q^{94} - 70 q^{95} - 15 q^{96} - 60 q^{97} - 4 q^{98} - 148 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9 - 9 * q^10 - 48 * q^11 - 15 * q^12 - 30 * q^13 - 29 * q^14 - 51 * q^15 + 69 * q^16 - 37 * q^17 + 40 * q^18 - 72 * q^19 - 9 * q^20 - 38 * q^21 - 48 * q^22 - 75 * q^23 - 15 * q^24 + 18 * q^25 - 30 * q^26 - 48 * q^27 - 29 * q^28 - 27 * q^29 - 51 * q^30 - 61 * q^31 + 69 * q^32 - 29 * q^33 - 37 * q^34 - 64 * q^35 + 40 * q^36 - 42 * q^37 - 72 * q^38 - 68 * q^39 - 9 * q^40 - 49 * q^41 - 38 * q^42 - 95 * q^43 - 48 * q^44 - 20 * q^45 - 75 * q^46 - 62 * q^47 - 15 * q^48 - 4 * q^49 + 18 * q^50 - 76 * q^51 - 30 * q^52 - 28 * q^53 - 48 * q^54 - 76 * q^55 - 29 * q^56 - 44 * q^57 - 27 * q^58 - 68 * q^59 - 51 * q^60 - 62 * q^61 - 61 * q^62 - 91 * q^63 + 69 * q^64 - 79 * q^65 - 29 * q^66 - 116 * q^67 - 37 * q^68 - 23 * q^69 - 64 * q^70 - 89 * q^71 + 40 * q^72 - 60 * q^73 - 42 * q^74 - 47 * q^75 - 72 * q^76 + 5 * q^77 - 68 * q^78 - 170 * q^79 - 9 * q^80 - 3 * q^81 - 49 * q^82 - 82 * q^83 - 38 * q^84 - 81 * q^85 - 95 * q^86 - 51 * q^87 - 48 * q^88 - 78 * q^89 - 20 * q^90 - 85 * q^91 - 75 * q^92 - 21 * q^93 - 62 * q^94 - 70 * q^95 - 15 * q^96 - 60 * q^97 - 4 * q^98 - 148 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more